Discrete Mathematics - Note

Exercise 8-11 [2] All S are V. Some V are not I. Some S are not I. Venn diagram: Two overlapping circles labeled S and V. The circle for I does not overlap with S. Conclusion: Invalid [3] All T are M. Some T are not S. Some M are not S. Venn diagram: Three overlapping circles labeled T, M, and S. The circle for T is completely contained within the circle for M. Conclusion: Valid [5] All voters are citizens. Some citizens are not residents. Some voters are not residents. Venn diagram: Three overlapping circles labeled V, C, and R. The circle for V is completely contained within the circle for C. Conclusion: Invalid [6] All D are M. No M are S. No S are D. Venn diagram: Three overlapping circles labeled D, M, and S. The circle for D is completely contained within the circle for M, and there is no overlap between M and S. Conclusion: Valid [8] All systems with unlimited storage capacity are systems with removable disks. No standard hard drives are systems with removable disks. No standard hard drives are systems with unlimited storage capacity. Venn diagram: Three overlapping circles labeled U, R, and H. The circle for U is completely contained within the circle for R. Conclusion: Valid Exercise 8-19 [2] All rats are disease-carrying pests. All disease-carrying pests are things that should be eradicated. All rats are things that should be eradicated. Venn diagram: Three overlapping circles labeled R, D, and E. The circle for R is completely contained within the circle for D, and the circle for D is completely contained within the circle for E. Conclusion: Valid [5] All fundamentalist Christians are evangelicals. All revivalists are evangelical. Some fundamentalist Christians are revivalists. Venn diagram: Three overlapping circles labeled F, R, and E. The circles for F and R are both completely contained within the circle for E. Conclusion: Invalid [7] No intramural sports are Olympic sports. Some intercollegiate sports are not Olympic sports. Some intercollegiate sports are intramural sports. Venn diagram: Three overlapping circles labeled IC, IM, and O. The circle for IM does not overlap with the circle for O. Conclusion: Invalid Exercise 8-19 (Continued) [8] All moars are Dinornithidae. No moars are things that exist. No Dinornithidae are things that exist. Venn diagram: Three overlapping circles labeled M, D, and E. The circle for M is completely contained within the circle for D, and there is no overlap between D and E. Conclusion: Invalid [13] No investor-held utilities are things that should be allowed to raise rates. All public utilities are things that should be allowed to raise rates. No public utilities are investor-held utilities. Venn diagram: Three overlapping circles labeled I, P, and R. The circle for P is completely contained within the circle for R, and there is no overlap between I and R. Conclusion: Invalid [21] All creationists are religious people. All fundamentalists are religious people. All creationists are fundamentalists. Venn diagram: Three overlapping circles labeled C, F, and R. The circles for C and F are both completely contained within the circle for R. Conclusion: Invalid Exercise [23] Statements: All people who voted for Democrats are people who favor expansion of medical services for the needy. All people who favor expansion of medical services for the needy are people who favor higher taxes. All people who voted for Democrats are people who favor higher taxes. Conclusion: All D are T. Venn diagram: Three overlapping circles labeled D, M, and T. The circle for D is completely contained within the circle for M, and the circle for M is completely contained within the circle for T. Validity: Valid